Solve By Using the Quadratic Equation Lessons

The quadratic formula gives us an alternative to Completing the
Square when we cannot factor an equation. People often find the
Quadratic Formula method easier and more convenient because it does not
require many operations on the equation being solved.

For solving an equation in the variable x, the Quadratic Formula is:

To find the solutions to an equation, we simply need to identify what
a, b, and c are, then substitute them into this formula, and simplify.

First Example (Two Solutions)

We begin applying the Quadratic Formula by putting the equation
in the following form:

Where a, b, and c are constants

This means that each term in the equation must be on the left side,
just like when we are factoring or Completing the Square. So we subtract
from each side.

Now for consistency, we will rearrange the terms so that they are
in the same order: The x2 term first, the x term second,
and the constant term last.

Now by comparing our equation with "ax2 + bx + c = 0",
we can see that a must equal 1, b must equal 1, and
c must equal .

Now that the values of a, b, and c have been determined, we may
return to the quadratic formula and use substitution. (Remember to
use parentheses when substituting to avoid problems with negative
signs.)

We must now simplify this equation keeping the Order of Operations
in mind. We begin by simplifying (1)2.

Next, we simplify multiplication. We see that 11 is equal to
15:

Now 1 and 15 are added, resulting in 16.

The square root of 16 is 4.

You may recall that the methods of solving by factoring and solving
by completing the square required you to split each problem into
multiple subproblems to obtain multiple solutions. Since we are again
looking for more than one solution, we must split this problem in two.

So far, we have been carrying the ?? sign through the problem.
Now, we will create two problems, one with a plus sign, and one with
a minus sign.

and

Simplifying the first subproblem gives

Simplifying the second subproblem gives

We can now combine these two solutions into the solution to the
original example problem: